Linear Algebra
Study Guide for Exam
3
Exam 3 is on Chapter 4. We covered 4.1 thru 4.7.
I will NOT give you the theorems on this exam. You need to know them.
Most
were introduced in chapter 2 and should be very familiar to you.
You may use a graphing c
alculator. The only types I will not allow are those that
do symbolic manipulations (e.g., TI

89).
You will get a take

home part of the exam.
Section
4.1
You need to know the definitions for or how to do
/use
the following:
Prove or disprove a given set
is a vector space. This will be on the take home.
Check that a given set is a subspace of a vector space.
The span of a set of vectors forms a subspace.
Section
4
.2
You need to know the definitions for or how to do
/use
the following:
Definition of
the nu
ll space.
The null space forms a subspace of R
n
.
Given a matrix A, find the null space explicitly.
Definition of the column space.
The column space of A is a subspace of R
m
.
Know the fact in the blue box on page 230.
The definition for a linear transformat
ion on page 232.
Know the definition for the kernel and range of a linear transformation. This is
just below the green box on page 232 or get it from your lecture notes.
Section
4
.3
You need to know the definitions for or how to do
/use
the following:
The
definition of a linearly independent set which is on page 237 (equation 1).
Know theorem 4.
Definition of a basis for a subspace of a vector space (also is the definition for the
basis of the vector space itself). See page 238.
The Spanning Set Theorem o
n page 239.
How to find a basis for Nul(A) and Col(A).
Read “Two Views of a Basis” on page 242.
Section
4
.4
You need to know the definitions for or how to do/use the following:
Know the content of the Unique Representation Theorem (page 246). The
coordin
ates of a vector with respect to a given basis are unique.
The definition on page 246; just be familiar with the notation.
Know how to find the change of coordinates matrix. See equation 4 on page 249.
Theorem 8.
Section
4
.5
You need to know the defin
itions for or how to do
/use
the following:
Be clear on the content of theorems 9 and 10.
Know the definition of dimension on page 257.
Know theorems 11 and 12.
Section
4
.6
You need to know the definitions for or how to do/use the following:
Know the defin
ition for the row space of A. See page 263 or your notes.
Theorem 13.
Know how to find a basis for
Row (
A) given a matrix A.
Know the definition of rank; page 265.
The Rank Theorem.
The new entries in the Invertible Matrix Theorem on page 267.
Section
4
.
7
You need to know the definitions for or how to do/use the following:
Know how to find the coordinates of a given vector with respect to different
bases.
Theorem 15.
Section
4.8 and 4
.9
We skipped 4.8 and 4.9.
To study for the exam, know the above list
of topics. Do your homework. Review your
quizzes. Do the supplementary exercises on pp.
298

300
(odds are in the back of the
book).
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